a martingale relative to a filtration
Saying a process is a martingale is incomplete until you say a martingale with respect to WHAT information. The crude version conditions on M_0, ..., M_n — the process's own past. But often you observe more than just the process: side information, other signals, the full record of the experiment. The amount of information available by time n is bundled into a sigma-algebra F_n, and the growing chain F_0 included in F_1 included in F_2 ... is a filtration — the formal model of 'knowledge accumulating over time, never forgotten'. A martingale is always a martingale relative to some specified filtration.
The full definition has three parts. First, the process must be adapted to the filtration: each M_n is F_n-measurable, meaning its value is knowable from the information available at time n (no peeking ahead). Second, each M_n is integrable, E[|M_n|] finite. Third, the fairness condition is stated against the filtration: E[M_(n+1) given F_n) = M_n. Conditioning on the richer F_n rather than just the process's own history makes the property genuinely stronger — it is harder to be a martingale with respect to a bigger flow of information, because your forecast must remain unmoved even when you know more. Whenever a process is a martingale relative to F_n, it is automatically a martingale relative to its own (smaller) generated filtration, but not conversely.
Why insist on this bookkeeping? Because stopping times, betting strategies, and the optional stopping theorem are all defined relative to the same filtration, and they only fit together when 'what you know at time n' is fixed once and for all. The classic trap: a process can be a martingale for one filtration and fail for a larger one. For example, future-revealing information can destroy fairness — if F_n secretly already 'knows' tomorrow's flip, the conditional expectation of M_(n+1) is no longer M_n. Always name the filtration; an unqualified 'martingale' implicitly means with respect to the natural filtration generated by the process itself.
Let X_1, X_2, ... be fair +/-1 steps and F_n the information from the first n steps. The walk S_n = X_1 + ... + X_n is a martingale relative to (F_n): S_n is F_n-measurable, integrable, and E[S_(n+1) given F_n] = S_n + E[X_(n+1)] = S_n. But relative to a filtration that already revealed X_(n+1), the same S_n would fail the fairness equation.
A filtration is the formal 'information so far'; the martingale property is stated relative to it.
Always name the filtration. A process can be a martingale for one and not for another; an unqualified 'martingale' means relative to its own natural filtration.